[petsc-users] newbie questions on preconditioner LU

[Matthew Knepley]

On Mon, Jan 28, 2013 at 2:27 PM, Ling Zou <lingzou80 at gmail.com> wrote:

>
>
> On Mon, Jan 28, 2013 at 12:01 PM, Matthew Knepley <knepley at gmail.com>wrote:
>
>> On Mon, Jan 28, 2013 at 1:39 PM, Ling Zou <lingzou80 at gmail.com> wrote:
>>
>>> Hi, All
>>>
>>> I am trying to understand how the preconditioner works when using KSP.
>>>
>>> For example, when using KSP to solve the linear system problem,
>>>
>>> Ax = b
>>>
>>> with the default left preconditioning. We actually solve,
>>>
>>> M^(-1) * A x = M^(-1) * b
>>>
>>> where, M is the preconditioning matrix and in many cases, we just use A
>>> as the preconditioning matrix.
>>>
>>>
>>> Question:
>>> 1), Is the understanding above correct?
>>>
>>
>> This is too simplistic. If you really mean M^{-1}, then no, you (almost)
>> never use A as M. If you mean an
>> approximate inverse to M, then yes.
>>
>>
>>> 2), If the understanding above is correct, is it correct to state the
>>> different methods provided in PETSc (such as PCLU, PCILU, etc) are to
>>> calculate the inverse matrix M^(-1) from M?
>>>
>>
>>  An approximate inverse.
>>
>>
>>> 3), How to understand this sentence in the manual (PETSc Users Manual,
>>> Reversion 3.3, page 78, under 4.4 Preconditioners)
>>> "The direct preconditioner, PCLU, is, in fact, a direct solver for the
>>> linear system that uses LU factorization. PCLU is included as a
>>> preconditioner so that PETSc has a consistent interface among direct and
>>> iterative linear solvers."
>>> Does this indicate when using PCLU, we solve Ax = b directly using LU
>>> factorization, or, we solve M^(-1) from M using LU factorization?
>>>
>>
>> Same thing, if M = A,
>>
>>   M^{-1} A x = A^{-1} A x = x = A^{-1} b
>>
>> which is Gaussian elimination for the original problem.
>>
>
> Ahh.. that's true!
> In case M is not A (as you pointed out earlier), does PCLU provide the
> approximated inverse matrix of M^{-1} using LU factorization on M?
>

Yes.

   Matt


>
>>    Matt
>>
>>
>>> As a beginner to the PETSc, all questions are probably too simple. I'd
>>> appreciate it if someone could answer my questions.
>>>
>>> Best,
>>>
>>> Ling
>>>
>>
>>
>>
>> --
>> What most experimenters take for granted before they begin their
>> experiments is infinitely more interesting than any results to which their
>> experiments lead.
>> -- Norbert Wiener
>>
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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